Thursday, 29 April 2010

Illustrating symmetry

On the assumption that I might need to explain spidrids in some detail, I thought I could use some diagrams of how radial symmetry works. In turn, that gave rise to the idea that bilateral symmetry could be used to contrast radial symmetry with. Perhaps no such explanations are necessary, but the images proved easy to produce, so here they are! I made rough 3D versions of a hexapod neocarnivore and of an eight-legged spidrid in ZBrush, and exported the result to Vue Infinite.

Click to enlarge; copyright Gert van Dijk

The image above shows a fairly robust hexapod as a good example of an animal with bilateral symmetry. The three images show a translucent plane bisecting the animal. The horizontal plane (top) and the vertical plane separating the front from the hind sides (bottom) show that the resulting two parts of the animal do not resemble one another. The middle image separates the left from the right sides of the animal, and these sides are mirror images of one another. The plane in the middle image, called a sagittal plane, is therefore a plane of symmetry dividing the animal into two mirrored halves. The two is translated as 'bi' and 'side' as lateral, hence 'bilateral'. The result of all this is that the top differs from the bottom, so it pays to distinguish the two. Likewise, front and back denote completely different aspects and hence functions. Only left and right are identical if mirrored. Easy, right? In real life bilateral symmetry should not be taken too far: internal organs can be quite asymmetrical, and organs with symmetrical external appearances may still show different functions for left and right sides, such as the human brain. But never mind that.

Click to enlarge; copyright Gert van Dijk

The next image shows a Furahan spidrid with radial symmetry. Its body contains eight equal segments -it is octomeric-, forming the body rather like slices of a pie form an entire pie. One such slice is shown in yellow, once in its normal position, and just for fun also as if one slice is shifted a bit, like people tend to do with pie charts. On Earth, there are quite a few radial animals. Starfish are a nice example. One of their odder characteristics is that their larvae show bilateral symmetry, suggesting that radial symmetry is a later development in these animals. While starfish have five segments, spidrids have eight, but the number does not really matter. Spidrids do have tops and bottoms, but what they emphatically do not have are front and rear sides, nor left and right sides. The terms simply do not apply; it might be better to speak of central and peripheral to distinguish which spot of the animal you are referring to.

Some of you may point out that the spidrid can be divided into two mirror halves using a plane, just like the neocarnivore. This would be absolutely true, but does not make the animals bilaterally symmetrical. The thing is that the resulting half would still contain four equal portions, so this way of dividing it does not go far enough. Using a plane of symmetry is not really valid to describe such an animal; it does not have a plane of symmetry but an axis of symmetry, running from the top to the bottom right through the centre of the animal.

Click to enlarge; copyright Gert van Dijk

Is there a minimum number of slices for radial symmetry? Theoretically there is no maximum, but the minimum number is intriguing. The red thingy in the image above shows an animal with three such segments, a state you might call 'trimerism'. I do not think any such scheme exists on Earth, and the results does not look at all like something I aim to have on Furaha. Perhaps someone can find a use for such a scheme as a floating life form hidden in plankton. The blue ridiculosity shows 'biradial symmetry'. You might think it has bilateral symmetry, but it doesn't: there still is no front or rear, nor left and right, to this beast. Rather than right it seems to be wrong. Still, believe it or not, 'biradial symmetry' exists! Just check Wikipedia. But do not expect anything with legs as shown here...

Click to enlarge; copyright Gert van Dijk

At this point all should seem clear, which is the right time to complicate matters. Going back to the spidrid, its eight slices can be shown up by cutting the animal up with four planes. The top image shows how this results in the sort of segment we started with. But the image below is equally valid, in that it too results in eight identical slices. Still, the slices are different. The way to reconcile this is to look closer at one segment on its own: it has bilateral symmetry with a plane of symmetry! Take care though: this does not hold for the animal as a whole, but for its slices. In fact, there are eight clockwise half segments and eight anticlockwise half segments. You could say that animals such as spidrids and starfishes do not exhibit perfect radial symmetry becuase of this, but that would take 'biological correctness' too far, I think...

j. w. bjerk said...

Nice illustrations!

"At this point all should seem clear, which is the right time to complicate matters..."
When i read that i was afraid you were going to delve into primary and secondary patterns of symmetry

Evan Black said...

"Perhaps someone can find a use for such a scheme as a floating life form hidden in plankton."

There are several nereid species that exhibit trimetric radial symmetry.